Abstract:The purpose of this article is to classify the real
hypersurfaces in complex space forms of dimension 2
that are
both Levi-flat and minimal. The main results are as
follows:
When the curvature of the complex space form is
nonzero, there is a 1-parameter family of such
hypersurfaces.
Specifically, for each one-parameter subgroup of the
isometry
group of the complex space form, there is an
essentially
unique example that is invariant under this
one-parameter
subgroup.
On the other hand, when the curvature of the space
form is zero, i.e., when the space form is complex
2-space
with its standard flat metric, there is an additional
`exceptional' example that has no continuous
symmetries but
is invariant under a lattice of translations. Up to
isometry
and homothety, this is the unique example with no
continuous
symmetries.